Computers rod. Engng Vol. 14, No. 2, pp. 103--112, 1988 0360-8352/88 $3.00+0.00 Printed in Great Britain Pergamon Press plc

A N E V A L U A T I O N O F S T A T I C F L O W S H O P S C H E D U L I N G H E U R I S T I C S IN D Y N A M I C F L O W S H O P M O D E L S V I A A

C O M P U T E R S I M U L A T I O N

YANG BYUNG PARK

Kuk Dong Apt. 3--702, Moon Jung Dong, Kang Dong Ku, Seoul 134, Korea

(Received for publication 19 August 1987)

Abstract--This paper provides an evaluation of static flowshop scheduling heuristics for minimiz- ing makespan as an objective function in the dynamic flowshop model. A total of 16 scheduling heuristics including several revisions and combinations of previously reported methods are summarized. The scheduling rules are evaluated via computer using a SLAM discrete event simulation model. The results for the simulation are analyzed using statistical methods. The results from the study suggest which of the popular scheduling heuristics hold promise for application to practical dynamic flowshop problems.

1. INTRODUCTION

A flowshop is a job shop on which several severe restrictions are placed. Here, the general static flowshop problem can be characterized with the following conditions:

(1) Each of N jobs is processed through M different machines in the same sequence. (2) Set-up times for the operations are sequence-independent and are included in

processing times. (3) Job descriptions such as processing times, set-up times, and processing methods

are known in advance. (4) M different machines are continuously available. (5) Individual operations are not preemptable. (6) A job does not become available to the next machine until it completes its work on

the current machine. Flowshop problems are classified into two categories: static models and dynamic

models. In the static situation, a set of operation jobs with deterministic processing times is available for processing at time zero. On the other hand, in the dynamic situation, jobs with stochastic processing times arrive at the shop randomly over time and are added into waiting jobs for processing. Therefore, job scheduling decisions must be made without knowledge of what and when jobs will arrive in the future.

Schedules are generally evaluated by aggregate quantities that involve information about all jobs, resulting in one-dimensional performance measures. These measures are usually expressed as a function of the set of job completion times. One of the most frequently used performance measures is makespan. Other important performance measures are mean flow time, mean job tardiness, total machine utilization, average in- process inventory, and average work content.

Little et al. [1], Lomnicki [2], and Bestwick and Hastings [3] applied branch-and- bound technique to determine the optimum job sequence for small static flowshop problems, with respect to makespan as an objective function. Littger [4] developed an algorithm for finding the optimal solution to the N x M static flowshop scheduling problems. The basis of the algorithm is to change the data of the problem iteratively while maintaining an optimal solution along the path. Baker and Schrage [5] introduced dynamic programming approach for finding the optimal sequence when jobs are related by precedence restrictions. However, each of these techniques has limitations to be employed in practical situations due to their computational requirements, so their applications have been constrained only to small flowshop problems.

Many good heuristics have been developed for the flowshop scheduling problems. Whenever heuristics have been proposed, simple and limited comparison tests have been

103

104 Y ~ c BYuNG PAItK

made to demonstrate the efficiency of the proposed algorithm. The studies performed to date have the common feature of the static situation where a dynamic and/or stochastic property is not considered that is frequently expected to exist in real situations.

In this paper, I survey static flowshop scheduhng heuristics and compare the perform- ance of these algorithms in a dynamic environment. The heuristics are evaluated on randomly generated dynamic flowshop problems via computer using a simulation language named SLAM (Simulation Language for Alternative Modeling).

2. H E U R I S T I C S F O R F L O W S H O P S C H E D U L I N G

During the past 20 yr, many heuristics have been developed to minimize makespan as an objective function. These heuristics can be divided into three categories:

(1) the application of Johnson's two-machine algorithm; (2) the generation of a slope index for the job processing times; (3) the minimization of total idle time on machines.

Table 1 gives a general description of heuristics mentioned in this research.

Table I. Genera l description of heuristics ment ioned in research

Heuristic name/ Basic idea of sequencing procedure The number identification of

Johnson 's algori thm Slope Total sequences index machine generated

Single Multiple idle t ime

!. P E T R O l x 2 2. P E T R O 2 x x 2 3. G U P T A x 1 4. R A P x x 1 5. RACS x x m - 1 6. C A M P I x m - 1 7. CAMP2 x x m - I 8. CAMP3 x m - I 9. P A L M E R x 1

10. BONNE1 x l 11. B O N N E 2 x x 1 12. SPAC1 x n 13. SPAC2 x n 14. G E L D E R x 1 15. N E H [n(n+ 1)/2]-1 16. D E L T A x 1

The heuristics chosen for testing came from both the literature and the ideas generated during analysis of the problems and methods. A total of 16 different heuristics have been selected for evaluation--12 from the literature and four variations proposed as part of this research. All of these heuristics will be described in the following sections.

Heuristics chosen from the literature.

(1) The Petrov heuristic (PETROl) V. A. Petrov [6] proposed a heuristic to generate a pair of not necessarily different

schedules, choosing the best schedule with regard to makespan. This heuristic is actually an extension of Johnson's algorithm for minimizing makespan in the two-machine flowshop.

(2) The Gupta heuristic (GUPTA) J. N. D. Gupta [7] argued that the sequencing problem is a problem of sorting n items

so as to minimize the makespan. He extended a sorting function for Johnson's two- and three-machine cases to an approximate function for the general m-machine case.

Static flowshop scheduling 105

(3) The Dannenbring heuristic (RAP)

D. G. Dannenbring [8] proposed a heuristic to provide a good solution quickly and easily. It is called the rapid access procedure (RAP), and it uses a weighting scheme similar to that for the slope order and Campbell et al. method. Dannenbring forms a single, two-machine sub-problem in which the processing times are transformed by the weighting scheme and the sub-problem is solved using Johnson's two-machine algorithm.

(4) The Dannenbring heuristic with close order search (RA CS)

D. G. Dannenbring [8] added a simple one-stage improvement process to his rapid access procedure. He defined (m-1) new sequences that can be formed by the transposi- tion of a single pair of adjacent neighbours in the rapid access solution. Then the best schedule with respect to makespan is chosen from among the (m-1) schedules.

(5) The Campbell, Dudek and Smith heuristic (CAMP1) The paper was presented by Campbell et al. [9], suggesting a new flowshop heuristic.

The authors treated the problem as ( m - l ) two-machine sub-problems and then con- structed ( m - 1) schedules using Johnson's algorithm. In their heuristic, a schedule is built by viewing total processing time as being made up of three parts: a fixed part in the centre with a variable part on each end. The heuristic is designed to minimize the two variable portions of the total time.

(6) The Palmer heuristic (PALMER)

D. S. Palmer [10] generalized the idea of using slopes as the basis of a sequencing rule by defining a numerical 'slope index' for a job with m operations. Then the job having the greatest slope is sequenced first, and so on. The major advantage of the Palmer heuristic is that, regardless of the number of machines, only one permutation schedule would be generated.

(7) The Bonney and Gundry heuristic (BONNE1)

M. C. Bonney and S. W. Gundry [11] devised a slope matching method using geo- metric relationships between the cumulative processing times. The shape of the job profiles is approximated by fitting linear regression lines to the start and end times of each operation, and then a job sequence is chosen by matching the start slope of one job with the end slope of the previous job.

(8) The Bonney and Gundry heuristic using Johnson's two-machine algorithm (BONNE2)

M. C. Bonney and S. W. Gundry [11] also employed Johnson's two-machine algor- ithm to match jobs in their slope matching method. Subject to the assumption that start and end slopes are a reasonable representation of the job profile, these slopes are used to derive an equivalent two-machine problem.

(9) The King and Spachis heuristic (SPA C1)

J. R. King and A. S. Spachis [12] presented five heuristics for flowshop scheduling problems and carried out comparative tests using simulation methods. Two of them, SPAC1 and SPAC2, were selected in this research on the basis of their performance.

King and Spachis argued that minimizing total machine idle time is equivalent to minimizing total makespan. Thus the heuristic was used to build up a single chain sequence: the next job is always selected so as to minimize total between-jobs delay, and that job is added to the end of the chain from the set of unscheduled jobs. Each job is considered in turn as the first job in the sequence in order to decide the best one.

(10) The King and Spachis heuristic using weighting factor (SPA C2) J. R. King and A. S. Spachis [12] also mentioned that delays on early machines might

have no ultimate effect on the makespan, whereas delays on later machines are much

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106 Y~o~o B r u N o PARK

more likely to have a 'knock-on' effect that would cause delay to be induced into the final machine. Therefore any delay on the last machine has a direct consequence in extending the makespan. Hence, in order to provide a penalty for between-jobs delay, a weighting factor is employed. The simplest factor and the one finally chosen is the sequence number of the machine.

(11) The Gelders and Sambandam heuristic (GELDER)

L. F. Gelders and N. Sambandam [13] presented a heuristic for minimizing a complex cost function and suggested the extension of this heuristic to the makespan problem. They used dynamic lateness and sequenced the jobs in such a way as to minimize idle time.

(12) The N E H heuristic (NEH)

M. Nawaz, E. Enscore and I. Ham [14] proposed a flowshop scheduling heuristic. They assumed that a job with a higher total processing time needs more attention than a job with a lower processing time. Because an exhaustive search technique is employed in this procedure, the number of enumerations in the whole process is large and is given by [n(n+ 1)/2]-1, where n equals the number of jobs.

In the first instance, only those two jobs with the highest total processing time are selected, and the best partial sequence for these two jobs is found by exhaustive search, i.e. trying the two possible combinations. The relative position of these jobs with respect to each other is fixed in the process which follows. Then the third job with the highest total processing time is selected from the remaining (n-2) jobs, partial sequences are tested, and the optimal sequence is found by placing this new job at all possible positions in the partial sequence found in the first step. This process is repeated until all jobs are fixed and a complete sequence is found.

Proposed Heuristics

(1) Combined Petrov and Palmer heuristic (PETR02)

The assumption underlying this combined heuristic is that the two components divided in the Petrov heuristic (PETROl) may be considered as two independent sub-problems having kl machines each. The Palmer's weighting factor may then be applied to the calculation of Petrov's artificial processing time in each sub-problem.

Hence two fictitious processing times for each of n jobs are newly defined as follows:

k, ( 2 j - k ~ - l ) til = Z 2 t,/

j = i i = 1 , 2 . . . n

m

ti2= ~ ( 2 j - 3 k ~ + l ) 2 ti; if m is odd j = k2

= ~ (2 j -3kt -1 ) 2 .ti/ if m is even

j = k2

where

kl =/¢z = [rrg2] + 1 if m is odd and k~ = [m/2],/% = [m/2] + 1 if m is even.

For sequencing the jobs, Johnson's two-machine algorithm and a decreasing ordering of (ti2 - til), i = 1,2 . . . . n are employed as in the Petrov heuristic (PETROl).

Static flowshop scheduling 107

(2) Combined Campbell et al. and Dannenbring heuristic (CAMP2)

This heuristic applies a weighting scheme for the Dannenbring's slope order to the Campbell et al. heuristic. The processing times for jobs are newly established as follows:

ttj k) = ~ (m - j + 1).t~/ i= 1 , 2 , . . . n

/'=1

k

t~ k) = E (m - j + l )'ti,m-j+ l k = l , 2 . . . . m - 1 ]=1

For each sub-problem, a job sequence is determined using Johnson's algorithm. The best with respect to makespan is selected from among the (m-1) sub-problems.

(3) Combined Campbell et al. and Page heuristic (CAMP3)

Page [15] showed that Johnson's two-machine algorithm consisted of two parts: calculating an address index for each job and then ordering the jobs on the basis of the address indices. This heuristic combines Page's address index for ordering a job sequence with the Campbell et al. procedure (CAMP1). Thus an address index for job i in sub- problem k of the Campbell et al. heuristic, g~i, is defined as follows:

g~ = sgn(t~ - t k )/min(t k , t k)

The job with the smallest index is assigned to the earliest sequence position, next smallest to the second position, etc.

(4) Heuristic using lower bound on makespan (DELTA)

The idea applied in this heuristic is that if the calculated lower bound on the makespan reasonably represents the real makespan, then minimizing the lower bound will be equivalent to minimizing the makespan.

Let o denote the set of jobs already scheduled, 0 the set of jobs still to be scheduled. Dim the delay time of job i on the last machine m when job i is added to o, and S(o,m) the finishing time of partial sequence o on the last machine. Then the lower bound on makespan, when job i is appended to the current partial schedule o, can be defined as:

Ti = S(o, m) + Di., + t~ + ~ tl,. I~0 I ~ i

It can be seen from the equation that the makespan is minimized by adding, from the set of unscheduled jobs, the job with the smallest delay time to the current partial schedule, because the other terms are constant. This procedure is repeated until all jobs have been scheduled.

3. COMPUTER SIMULATION FOR DYNAMIC FLOWSHOP MODEL

In the dynamic flowshop situation, jobs with stochastic processing times arrive at the shop randomly over time, and are added into the waiting jobs for processing.

Clearly, an ideal solution in the dynamic model would be to reschedule the waiting jobs with a best method as each new job arrives. But as this is not generally practical from a computational or theoretical viewpoint, rescheduling has to be carried out less frequently. The actual frequency of this rescheduling is clearly of importance, and some of static flowshop scheduling heuristics may be more adaptable to the dynamic case than to the static case. The objective is to develop a simulation model that can be used to

108 YANG BYUNG P,~.K

analyz.e the performance-effectiveness of different heuristics with different rescheduling frequencies on the dynamic flowshop.

The conditions for constructing a discrete event dynamic flowshop simulation model are arranged as follows:

(1) jobs are processed through 8 machines which are set to idle initially; (2) a set of 10 jobs is initially waiting for processing at the first machine; (3) a total of 100 jobs are scheduled to arrive at the shop randomly over time with an

interval following a uniform distribution with an interval of 11 and 25 time units; (4) each job processing time has an exponential distribution with a mean of a sample

randomly generated from an Erlang distribution which is the sum of 10 exponential samples each with a mean of 2;

(5) a sequence for the waiting jobs is repeatedly scheduled by a heuristic every fixed number of time units. Erling distribution was employed for the job processing times because it is a low variance distribution which reduces the influence of random error on the performance of the heuristics.

SLAM developed by Pegden [16], based on earlier developments by Pritsker [16], is designed to provide a useful tool for simulation models when they are represented as network models, discrete event models, combined network-discrete event models, continuous models, or combined network-discrete event-continuous models. Five event subroutines are defined to simulate the discrete event dynamic flowshop model using SLAM as follows:

EVENT JARV: EVENT FMAC: EVENT AMAC: EVENT ENDM:

job arrival event at the shop, job arrival event at the first machine, job arrival event at all the machines but the first one, end-of-service event for the job being processed on a particular machine,

EVENT JSEQ: event for constructing a job sequence by a heuristic. Each discrete event is coded as a FORTRAN subroutine.

The organization of SLAM program for the dynamic model is depicted in Fig. 1. SLAM processor controls two-additional user-written subroutines besides the five event subroutines: INTLC and OTPUT. Subroutine INTLC is called by SLAM before each

I Main ]

, _ H I i statement Library in i t ia l izat ion p r o c e s s o r L

s,,0roo.,o. o0ro".,n. l I INTLC EVENT (1) OTPUT reports

6 A i I I I I l I I * |

I f I I I I i

SLAM Library of s u b p r o g r a m s

Fig. 1

Static f lowshop scheduling 109

simulation run and is used to set initial conditions and to schedule initial events. Subroutine OTPUT is called at the end of each simulation and is used for end-of- simulation processing such as printing problem specific results for the simulation.

4. E V A L U A T I O N O F T H E H E U R I S T I C S

4.1. Steel's rank sum test

Before the evaluation of performance-effectiveness of the static flowshop scheduling heuristics in the dynamic flowshop model, Steel's rank sum test was performed to analyze the performance differences of the heuristics with respect to makespan. The test was done on 1500 randomly generated static flowshop problems ranging from 3 x 4 to 30 x 20 problems. To support Steel's test, goodness-of-fit test was employed at the error rate 0.05 level.

Table 2 gives the results showing that there are significant differences in makespans due to the heuristic factor at the 5% experiment-wise error rate in all problems except for 3-job problems. Maximum rank sum statistics (R) is achieved by PETRO2 or BONNE1. Hence, it is guessed that PETRO2 or BONNE1 would be one of the worst heuristics in performance. Rank sum statistics are increased as the problem size is increased. This means that the differences in makespans among heuristics are significantly increased as the problem size is increased.

The fact that 3-job problems do not have significantly different makespans in a performance at the 5% error rate, can be explained in two ways: first, since there are only six feasible permutation schedules in 3-job problems, there is a high probability of

Table 2. Steel's rank sum test for the static flowshop problems (ct = 0.05)

Problem size (N x /14) Sample size R (i*)t Ho

3 x 4 50 484(2) A 8 50 489(10) A

12 50 484(10) A 16 50 490(10) A 20 50 487(10) A

6 x 4 50 548(2) R 8 50 54o(2) R

12 50 54o(1o) g 16 50 542 (10) R 20 50 570(2) S

9 × 4 50 566(2) R 8 50 574(2) R

12 50 577(2) R 16 50 588(2) R 20 50 602(10) R

15 x 4 20 593(2) R S 20 610 (2) R

12 20 603 (2) R 16 20 605(2) R 20 20 610 (2) R

2 0 x 4 20 597(2) R 8 20 610 (2) R

12 20 610 (2) R 16 20 610 (2) R 20 20 610 (2) R

3 0 x 4 20 608(2) R 8 20 610 (2) R

12 20 610 (2) R 16 20 610 (2) R 20 20 610 (2) R

? R denotes the maximum rank sum and i" denotes the heuristic number (See Table 1) for which the maximum rank sum was achieved. Ho: Pl = ~ • • • z Pie A: Accepted Hi: It1 * Pt, i ,- 2,3 . . . . . 16 R: Rejected

110

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8

E E

=0

o~

~ E

0~.

CG

ua

0 . 0

i+ .j

Static flowshop 'scheduling 111

obtaining an optimal solution. Second, the job processing times are generated by a low variance distribution which reduces the influence of random error on the performance of heuristics.

4.2. Common comparison test

For evaluation of performance-effectiveness of heuristics in the dynamic environment, the common comparison test was employed. Table 3 shows the results of the test for the heuristics in both static and dynamic models. The evaluation of the heuristics in the static flowshop models is based on the Park et al. report [17].

It is clear from the table that NEH is the least biased and the best-operated of the 16 heuristics tested with 'respect to makespan in both models. This means that NEH is the most adaptable to the dynamic situation as well as to the static situation. Although SPAC1 is the sixth best heuristic for the static model, it moves to the second best for the dynamic model. This shift is likely due to its flexibility in the dynamic situation. PETRO2, GUPTA, RAP, PALMER, and BONNE2 also show the improved rankings in the dynamic model. On the other hand, CAMP1 shows the largest drop in position. PETROl , SPAC2, and GELDER also generate considerably worse rankings in the dynamic case.

It is shown that the heuristics produce different effectiveness for a particular schedul- ing frequency. This is because jobs are not selected in order of their position in the sequence. In the high frequencies of 40 and 80 time units, only the first few jobs in the sequence are processed before the new schedule is constructed. On the other hand, in the low frequencies of 200, 240, and 280 time units, there may be unsequenced jobs between scheduling times. That is, many unsequenced jobs waiting in the queue are processed with a first-in first-out dispatching rule after the jobs already sequenced have been processed. This may explain why PALMER is the most effective heuristic in the low scheduling frequencies of 40 and 80 time units, while NEH is the best one in the other five scheduling frequencies.

In the simulation, PALMER produced the lowest mean makespan for the dynamic flowshop model constructed when jobs are rescheduled every 40 time units. In dynamic flowshop scheduling problems, it is clearly of importance to select a best rule together with a rescheduling frequency.

Besides a makespan, the simulation provided statistics about two other measures of performance-effectiveness in the dynamic model: the average in-process inventory in the shop and the average machine utilization. The information of the in-process inventory serves a good reference for the design of buffer size and the calculated machine utilization gives data for the machine load. Finally, it was confirmed through the simulation that the differential among heuristics in a performance is increased when the processing times with less variance are used, and vice versa.

5. CONCLUSION

In dynamic flowshop model, jobs with stochastic processing times arrive at the shop randomly over time and are added into the waiting jobs for processing. The major purpose of this paper was to evaluate the performance of static flowshop scheduling heuristics in the dynamic flowshop model using a SLAM discrete event simulation. The research resulted in the following conclusions:

(1) Steel's rank sum test showed that there are significant differences in makespans due to heuristics at the 0.05 error rate;

(2) the differential among heuristics in a performance was dependent on the variance of the job processing times;

(3) the frequency of rescheduling the jobs was an important factor to the performance of heuristics in dynamic situation;

(4) the results of dynamic flowshop simulation provided some valuable information regarding such factors as the buffer size and the machine load in the shop;

112 YANG BYUNO PARK

(5) some heuristics were more adaptable to dynamic model than to static model. In the dynamic flowshop constructed, in general, NEH was the most effective of the 16 followed by SPAC1, CAMP2, and CAMP3 heuristics while PALMER was the best one in the frequencies of 40 and 80 time units.

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